by Brian Greene
Why can't we figure out which is the "right" Calabi-Yau shape? Most string theorists blame this on the inadequacy of the theoretical tools currently being used to analyze string theory. As we shall discuss in some detail in Chapter 12, the mathematical framework of string theory is so complicated that physicists have been able to perform only approximate calculations through a formalism known as perturbation theory. In this approximation scheme, each possible Calabi-Yau shape appears to be on equal footing with every other; none is fundamentally singled out by the equations. And since the physical consequences of string theory depend sensitively on the precise form of the curled-up dimensions, without the ability to select one Calabi-Yau space from the many, no definitive experimentally testable conclusions can be drawn. A driving force behind present-day research is to develop theoretical methods that transcend the approximate approach in the hope that, among other benefits, we will be led to a unique Calabi-Yau shape for the extra dimensions. We will discuss progress along these lines in Chapter 13.
Exhausting Possibilities
So you might ask: Even though we can't as yet figure out which Calabi-Yau shape string theory selects, does any choice yield physical properties that agree with what we observe? In other words, if we were to work out the corresponding physical properties associated with each and every CalabiYau shape and collect them in a giant catalog, would we find any that match reality? This is an important question, but, for two main reasons, it is also a hard one to answer fully.
A sensible start is to focus only on those Calabi-Yau shapes that yield three families. This cuts down the list of viable choices considerably, although many still remain. In fact, notice that we can deform a multihandled doughnut from one shape to a slew of others—an infinite variety, in fact—without changing the number of holes it contains. In Figure 9.2 we illustrate one such deformation of the bottom shape from Figure 9.1. In much the same way, we can start with a three-holed Calabi-Yau space and smoothly deform its shape without changing the number of holes, again through what amounts to an infinite sequence of shapes. (When we mentioned earlier that there were tens of thousands of Calabi-Yau shapes, we were already grouping together all those shapes that can be changed into one another by such smooth deformations, and we were counting the whole group as one Calabi-Yau space.) The problem is that the detailed physical properties of string vibrations, their masses and their response to forces, are very much affected by such detailed changes in shape, but once again, we have no means of selecting one possibility over any other. And no matter how many graduate students physics professors might set to work, it's just not possible to figure out the physics corresponding to an infinite list of different shapes.
This realization has led string theorists to examine the physics resulting from a sample of possible Calabi-Yau shapes. Even here, however, life is not completely smooth sailing. The approximate equations that string theorists currently use are not powerful enough to work out the resulting physics fully for any given choice of Calabi-Yau shape. They can take us a long way toward understanding, in the sense of a ballpark estimate, the properties of the string vibrations that we hope will align with the particles we observe. But precise and definitive physical conclusions, such as the mass of the electron or the strength of the weak force, require equations that are far more exact than the present approximate framework. Recall from Chapter 6—and the Price is Right example—that the "natural" energy scale of string theory is the Planck energy, and it is only through extremely delicate cancellations that string theory yields vibrational patterns with masses in the vicinity of those of the known matter and force particles. Delicate cancellations require precise calculations because even small errors have a profound impact on accuracy. As we will discuss in Chapter 12, during the mid-1990s physicists have made significant progress toward transcending the present approximate equations, although there is still far to go.
So where do we stand? Well, even with the stumbling block of having no fundamental criteria by which to select one Calabi-Yau shape over any other, as well as not having all the theoretical tools necessary to extract the observable consequences of such a choice fully, we can still ask whether any of the choices in the Calabi-Yau catalog gives rise to a world that is in even rough agreement with observation. The answer to this question is quite encouraging. Although most of the entries in the Calabi-Yau catalog yield observable consequences significantly different from our world (different numbers of particle families, different number and types of fundamental forces, among other substantial deviations), a few entries in the catalog yield physics that is qualitatively close to what we actually observe. That is, there are examples of Calabi-Yau spaces that, when chosen for the curled-up dimensions required by string theory, give rise to string vibrations that are closely akin to the particles of the standard model. And, of prime importance, string theory successfully stitches the gravitational force into this quantum-mechanical framework.
With our present level of understanding, this situation is the best we could have hoped for. If many of the Calabi-Yau shapes were in rough agreement with experiment, the link between a specific choice and the physics we observe would be less compelling. Many choices could fit the bill and hence none would appear to be singled out, even from an experimental perspective. On the other hand, if none of the Calabi-Yau shapes came even remotely close to yielding observed physical properties, it would seem that string theory; although a beautiful theoretical framework, could have no relevance for our universe. Finding a small number of Calabi-Yau shapes that, with our present, fairly coarse ability to determine detailed physical implications, appear to be well within the ballpark of acceptability is an extremely encouraging outcome.
Explaining the elementary matter and force particle properties would be among the greatest—if not the greatest—of scientific achievements. Nevertheless, you might ask whether there are any string theoretic predictions—as opposed to postdictions—that experimental physicists could attempt to confirm, either now or in the foreseeable future. There are.
Superparticles
The theoretical hurdles currently preventing us from extracting detailed string predictions force us to search for generic, rather than specific, aspects of a universe consisting of strings. Generic in this context refers to characteristics that are so fundamental to string theory that they are fairly insensitive to, if not completely independent of, those detailed properties of the theory that are now beyond our theoretical purview. Such characteristics can be discussed with confidence, even with an incomplete understanding of the full theory In subsequent chapters we shall return to other examples, but for now we focus on one: supersymmetry.
As we have discussed, a fundamental property of string theory is that it is highly symmetric, incorporating not only intuitive symmetry principles but respecting, as well, the maximal mathematical extension of these principles, supersymmetry. This means, as discussed in Chapter 7, that patterns of string vibrations come in pairs—superpartner pairs—differing from each other by a half unit of spin. If string theory is right, then some of the string vibrations will correspond to the known elementary particles. And due to the supersymmetric pairing, string theory makes the prediction that each such known particle will have a superpartner. We can determine the force charges that each of these superpartner particles should carry, but we do not currently have the ability to predict their masses. Even so, the prediction that superpartners exist is a generic feature of string theory; it is a property of string theory that is true, independent of those aspects of the theory we haven't yet figured out.
No superpartners of the known elementary particles have ever been observed. This might mean that they do not exist and that string theory is wrong. But many particle physicists feel that it means that the superpartners are very heavy and are thus beyond our current capacity to observe experimentally. Physicists are now constructing a mammoth accelerator in Geneva, Switzerland, called the Large Hadron Collider. Hopes run high that this machine
will be powerful enough to find the superpartner particles. The accelerator should be ready for operation before 2010, and shortly thereafter supersymmetry may be confirmed experimentally. As Schwarz has said, "Supersymmetry ought to be discovered before too long. And when that happens, it's going to be dramatic."17
You should bear in mind two things, though. Even if superpartner particles are found, this fact alone will not establish that string theory is correct. As we have seen, although supersymmetry was discovered by studying string theory, it has also been successfully incorporated into point-particle theories and is therefore not unique to its stringy origins. Conversely, even if superpartner particles are not found by the Large Hadron Collider, this fact alone will not rule out string theory, since it might be that the superpartners are so heavy that they are beyond the reach of this machine as well.
Having said this, if in fact the superpartner particles are found, it will most definitely be strong and exciting circumstantial evidence for string theory.
Fractionally Charged Particles
Another experimental signature of string theory, having to do with electric charge, is somewhat less generic than superpartner particles but equally dramatic. The elementary particles of the standard model have a very limited assortment of electric charges: The quarks and antiquarks have electric charges of one-third or two-thirds, and their negatives, while the other particles have electric charges of zero, one, or negative one. Combinations of these particles account for all known matter in the universe. In string theory, however, it is possible for there to be resonant vibrational patterns corresponding to particles of significantly different electric charges. For instance, the electric charge of a particle can take on exotic fractional values such as 1/5, 1/11, 1/13, or 1/53, among a variety of other possibilities. These unusual charges can arise if the curled-up dimensions have a certain geometrical property: Holes with the peculiar property that strings encircling them can disentangle themselves only by wrapping around a specified number of times.18 The details are not particularly important, but it turns out that the number of windings required to get disentangled manifests itself in the allowed patterns of vibration by determining the denominator of the fractional charges.
Some Calabi-Yau shapes have this geometrical property while others do not, and for this reason the possibility of unusual electric-charge fractions is not as generic as the existence of superpartner particles. On the other hand, whereas the prediction of superpartners is not a unique property of string theory, decades of experience have shown that there is no compelling reason for such exotic electric-charge fractions to exist in any point-particle theory. They can be forced into a point-particle theory, but doing so would be as natural as the proverbial bull in a china shop. Their possible emergence from simple geometrical properties that the extra dimensions can have makes these unusual electric charges a natural experimental signature for string theory.
As with the situation with superpartners, no such exotically charged particles have ever been observed, and our understanding of string theory does not allow for a definitive prediction of their masses should the extra dimensions have the correct properties to generate them. One explanation for not seeing them, again, is that if they do exist, their masses must be beyond our present technological means—in fact, it is likely that their masses would be on the order of the Planck mass. But should a future experiment come across such exotic electric charges, it would constitute very strong evidence for string theory.
Some Longer Shots
There are yet other ways in which evidence for string theory might be found. For example, Witten has pointed out the long-shot possibility that astronomers might one day see a direct signature of string theory in the data they collect from observing the heavens. As encountered in Chapter 6, the size of a string is typically the Planck length, but strings that are more energetic can grow substantially larger. The energy of the big bang, in fact, would have been high enough to produce a few macroscopically large strings that, through cosmic expansion, might have grown to astronomical scales. We can imagine that now or sometime in the future, a string of this sort might sweep across the night sky, leaving an unmistakable and measurable imprint on data collected by astronomers (such as a small shift in the cosmic microwave background temperature; see Chapter 14). As Witten says, "Although somewhat fanciful, this is my favorite scenario for confirming string theory as nothing would settle the issue quite as dramatically as seeing a string in a telescope."19
Closer to earth, there are other possible experimental signatures of string theory that have been put forward. Here are five examples. First, in Table 1.1 we noted that we do not know if neutrinos are just very light, or if in fact they are exactly massless. According to the standard model, they are massless, but not for any particularly deep reason. A challenge to string theory is to provide a compelling explanation of present and future neutrino data, especially if experiments ultimately show that neutrinos do have a tiny but nonzero mass. Second, there are certain hypothetical processes that are forbidden by the standard model, but that may be allowed by string theory. Among these are the possible disintegration of the proton (don't worry, such disintegration, if true, would happen very slowly) and the possible transmutations and decays of various combinations of quarks, in violation of certain long-established properties of point-particle quantum field theory.20 These kinds of processes are especially interesting because their absence from conventional theory makes them sensitive signals of physics that cannot be accounted for without invoking new theoretical principles. If observed, any one of these processes would provide fertile ground for string theory to offer an explanation. Third, for certain choices of the Calabi-Yau shape there are particular patterns of string vibration that can effectively contribute new, tiny, long-range force fields. Should the effects of any such new forces be discovered, they might well reflect some of the new physics of string theory. Fourth, as we note in the next chapter, astronomers have collected evidence that our galaxy and possibly the whole of the universe is immersed in a bath of dark matter, the identity of which has yet to be determined. Through its many possible patterns of resonant vibration, string theory suggests a number of candidates for the dark matter; the verdict on these candidates must await future experimental results establishing the detailed properties of the dark matter.
And finally, a fifth possible means of connecting string theory to observations involves the cosmological constant—remember, as discussed in Chapter 3, this is the modification Einstein temporarily imposed on his original equations of general relativity to ensure a static universe. Although the subsequent discovery that the universe is expanding led Einstein to retract the modification, physicists have since realized that there is no explanation for why the cosmological constant should be zero. In fact, the cosmological constant can be interpreted as a kind of overall energy stored in the vacuum of space, and hence its value should be theoretically calculable and experimentally measurable. But, to date, such calculations and measurements lead to a colossal mismatch: Observations show that the cosmological constant is either zero (as Einstein ultimately suggested) or quite small; calculations indicate that quantum-mechanical fluctuations in the vacuum of empty space tend to generate a nonzero cosmological constant whose value is some 120 orders of magnitude (a 1 followed by 120 zeros) larger than experiment allows! This presents a wonderful challenge and opportunity for string theorists: Can calculations in string theory improve on this mismatch and explain why the cosmological constant is zero, or if experiments do ultimately establish that its value is small but nonzero, can string theory provide an explanation? Should string theorists be able to rise to this challenge—as yet they have not—it would provide a compelling piece of evidence in support of the theory.
An Appraisal
The history of physics is filled with ideas that when first presented seemed completely untestable but, through various unforeseen developments, were ultimately brought within the realm of experimental veri
fiability. The notion that matter is made of atoms, Pauli's hypothesis that there are ghostly neutrino particles, and the possibility that the heavens are dotted with neutron stars and black holes are three prominent ideas of precisely this sort—ideas that we now embrace fully but that, at their inception, seemed more like musings of science fiction than aspects of science fact.
The motivation for introducing string theory is at least as compelling as any of these three ideas—in fact, string theory has been hailed as the most important and exciting development in theoretical physics since the discovery of quantum mechanics. This comparison is particularly apt because the history of quantum mechanics teaches us that revolutions in physics can easily take many decades to reach maturity. And compared to today's string theorists, the physicists working out quantum mechanics had a great advantage: Quantum mechanics, even when only partially formulated, could make direct contact with experimental results. Even so, it took close to 30 years for the logical structure of quantum mechanics to be worked out, and about another 20 years to incorporate special relativity fully into the theory. We are now incorporating general relativity, a far more challenging task, and, moreover, one that makes contact with experiment much more difficult. Unlike those who worked out quantum theory, today's string theorists do not have the shining light of nature—through detailed experimental results—to guide them from one step to the next.
